The four-loop dilaton β-function

We employ the identity derived by Curci and Paffuti relating the metric and dilaton β-functions for the bosonic σ-model to deduce the O(α′³) (four-loop) dilaton β-function from the O(α′³) (three-loop) metric β-function. We also give details of an explicit four-loop calculation in partial confirmation of our result.     

Problems with the ζ-function method

The -function method for evaluating effective actions is applied to an operator containing a potential localized on two parallel planes. This operator is characterized by a continuous spectrum and broken translation invariance. In this case, the -method leads to a divergent volume factor independent of the physical parameters. In a suitable regularization scheme, only the next to leading order term reproduces a physically interesting result.     

Feynman propagator for the δ-function potential

A new derivation of the Feynman propagator for the Dirac δ-function potential is presented.     

β-function approach to the Green-Schwarz superstring

We describe the calculation of one-loop β-functions for the Green-Schwarz heterotic superstring in a supergravity background. The calculation is considerably simplified by the use of a new set of superspace constraints. As expected, the one-loop β-functions vanish, since the constraints impose on the background the classical field equations of supergravity.     

A ζ-function method for weyl fermionic determinants

We present an extension of the -function method adapted to handle the regularization of Dirac operator determinants when Weyl fermions are present. The method we propose makes use of an auxiliary operator which takes into account regularization ambiguities in anomalous gauge theories. As an application, we consider a two-dimensional model where these ambiguities allow for the definition of a consistent quantum theory.     

On ruelle's ξ-function for exponentially decreasing potentials

We show that Ruelle's generalised -function for a classical one-dimensional lattice spin system with two-body interaction (i) exp(-i )a(i) with >1 extends to a meromorphic function in the whole complex plane.     

A generalization of the one-dimensional δ-function Bose gas

The one-dimensional boson gas with -function interaction is modified to include arbitrary pseudopotential interaction. The system is shown to be solvable by the Bethe Ansatz for certain classes of pseudopotential.     

Three-loop β-function for the bosonic non-linear sigma model

We present a calculation of the three-loop metric β-function for the bosonic non-linear sigma model. We also construct the action whose equations of motion coincide with the vanishing of the β-function. It is found to be the effective action derived from string scattering amplitudes.     

Two-loop β-function for the generalized bosonic sigma model

We independently compute: (i) the order α′ terms in the effective action for the massless fields in the Bose string theory and (ii) the two-loop β-function in the corresponding two-dimensional σ-model. It is proved that the “massless string fields equations of motion equal the σ-model Weyl invariance conditions” conjecture is true to the given order     

Nilpotency ofQBRST and the vanishing ß-function in string theories

We study the conformal Ward identities following from the BRST invariance in string theories and show that the nilpotency of Q_BRST implies a vanishing ß-function and vice versa.     

One-dimensional δ-function attractive Fermi gas in an external magnetic field

We investigate the ground-state properties of a one-dimensional spin-1/2 fermionic atoms interacting through the attractive δ-function potential in an external magnetic field. By the thermal Bethe ansatz method, the integral equations for the dressed energies are formulated. Those integral equations at zero-temperature are solved in the power series forms of rapidity and momentum for both strong and weak coupling cases. The magnetization as a function of the coupling constant and the external field is also obtained explicitly. Based on the analytic results, quantum phase transitions are identified among three phases, an unpolarized fully paired state, a fully polarized normal ferromagnetic state and a mixed state of paired and unpaired atoms.     

Asymptotics of a τ-function arising in the two-dimensional Ising model

The short-distance assymptotics of the -function associated to the 2-point function of the two-dimensional Ising model is computed as a function of the integration constant defined from the long-distance behavior of the -function. The result is expressible in terms of the Barnes double gamma function (equivalently, the BarnesG-function).Supported in part by the National Science Foundation, Grant No. DMS-90-01794     

On the ζ-function of a one-dimensional classical system of hard-rods

The -function of a one-dimensional classical hard-rod system with exponential pair interaction is defined as the generating function for the partition function of the system with periodic boundary conditions. It is shown, here, that the -function for this system is simply related to the traces of the restrictions of the Ruelle's transfer matrix, and related operators to a suitable function space. This -function does not, in general, extend to a meromorphic function.     

Resonances and antibound states for the dressed one-dimensional δ-function potential

We calculate resonances, antibound states and two new light induced states of the dressed one-dimensional attractive δ-function potential, as function of α₀ — the free electron quiver motion amplitude in a monochromatic electromagnetic field, in the range 0<α₀<150 a.u. We describe also our method, adequate to the numerical solution of the Schrödinger equation for this particular potential.     

Green's function formulation of multiple nonlinear Dirac δ-function potential in one dimension

In this work, we study the scattering problem of the general nonlinear finitely many Dirac delta potentials with complex coupling constants (or opacities in the context of optics) using the Green's function method and then find the bound state energies and the wave functions for the particular form of the nonlinearity in the case of positive real coupling constants.     

Ground-state energy density of a one-dimensional fermion system with repulsive δ-function interaction: A pseudo-linear response treatment

An analytical treatment for the ground-state energy density of the system of one-dimensional fermions interacting via repulsive δ-function potential is presented. The treatment rests on a direct approximation to the pair distribution function and coupling constant integration. The analytical results, obtained for arbitrary coupling strength values, are in good agreement with those given by the numerical solution of an exact integral equation.